Numerical Analysis - Proof of the Linear Convergence of the Fixed-Point Iterative Method

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This video is the proof of the Linear Convergence of the Fixed-Point Iterative Method (Theorem) from Numerical Analysis.
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Hi, how do you get the result at 6:45 ? The result: There is a small neighborhood around r for which |g'(x)| < (S+1)/2. Thanks!

taoming